In this paper, we study the restriction of an irreducible unitary representation π of the universal covering S̃p2n(R) to a Heisenberg maximal parabolic group P̃ . We prove that if π|P̃ is irreducible, then π must be a highest weight module or a lowest weight module. This is in sharp constrast with the GLn(R) case. In addition, we show that for a unitary highest or lowest weight module, π|P̃ decompo...

Theta correspondence θ over R was established by Howe in ( [5]). In [10], we proved that θ preserves unitarity under certain restrictions, generalizing the result of Jian-Shu Li ( [16]). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual p...

Let 5 and 8 be Krein spaces. By S(3 ,8 ) we denote the (generalized Schur) class of all functions O, defined and holomorphic at z = 0 and with values in L(5,8), the space of bounded Linear operators from 5 to 8 (we write L ( 8 ) for L(3,S)). It is known, see Section 3 below, that any OeS(3,8) is in some neighborhood of z = 0 the characteristic function On of a unitary colligation A = (J?,S,@;U)...

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension 2 N−1 2 , and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian q...

Let G be a reductive Lie group and O the coadjoint orbit of a hyperbolic element of g. By π is denoted the unitary irreducible representation of G associated with O by the orbit method. We give geometric interpretations in terms of concepts related to O of the constant π(g), for g ∈ Z(G). In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differentia...

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted Lp-spa...

We use some Lie group theory and Budney’s unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on ...

Given an integer n, one defines the unitary Cayley graph, denoted Cay(Zn,Zn), to be the graph whose vertex set is Zn, the integers modulo n, with an edge between two vertices x, y if x− y is a unit in (the ring) Zn. Unitary Cayley graphs have been studied as objects of independent interest (see, for example, [3], [2], [7], [8], [9]) but are of particular relevance in the study of graph represen...

Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual Ĝ. The moment map for π ∈ Ĝ sends smooth vectors in the representation space of π to g∗. The closure of the image of the moment map for π is called its moment set. N. Wildberger has proved that the moment set for π coincides with the closure of the convex hull of the corresponding coadjoint orbit. ...

Gelfand-Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand-Graev characters of the finite unitary group from a combinatorial point of view. In particular, ...